Correlation and Linear Regression

Chapter 13

McGraw-Hill/Irwin Copyright © 2012 by The McGraw-Hill Companies, Inc. All rights reserved.

Learning ObjectivesLO1 Define the terms dependent and independent variable.

LO2 Calculate, test, and interpret the relationship between two

variables using the correlation coefficient.

LO3 Apply regression analysis to estimate the linear relationship

between two variables

LO4 Interpret the regression analysis.

LO5 Evaluate the significance of the slope of the regression equation.

LO6 Evaluate a regression equation to predict the dependent variable.

LO7 Calculate and interpret the coefficient of determination.

LO8 Calculate and interpret confidence and prediction intervals.

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Regression Analysis - Introduction

Recall in chapter 4 we used Applewood Auto Group data to show the relationship between two variables using a scatter diagram. The profit for each vehicle sold and the age of the buyer were plotted on an XY graph

The graph showed that as the age of the buyer increased, the profit for each vehicle also increased

This idea is explored further here. Numerical measures to express the strength of relationship between two variables are developed.

In addition, an equation is used to express the relationship between variables, allowing us to estimate one variable on the basis of another.

EXAMPLESDoes the amount Healthtex spends per month on training its sales force affect its monthly sales?

Is the number of square feet in a home related to the cost to heat the home in January?

In a study of fuel efficiency, is there a relationship between miles per gallon and the weight of a car?

Does the number of hours that students studied for an exam influence the exam score?

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Dependent Versus Independent Variable

The Dependent Variable is the variable being predicted or estimated.The Independent Variable provides the basis for estimation. It is the predictor variable.

Which in the questions below are the dependent and independent variables?

1.Does the amount Healthtex spends per month on training its sales force affect its monthly sales?2.Is the number of square feet in a home related to the cost to heat the home in January?3.In a study of fuel efficiency, is there a relationship between miles per gallon and the weight of a car?4.Does the number of hours that students studied for an exam influence the exam score?

LO1 Define the terms dependent and independent variable.

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Scatter Diagram Example

The sales manager of Copier Sales of America, which has a large sales force throughout the United States and Canada, wants to determine whether there is a relationship between the number of sales calls made in a month and the number of copiers sold that month. The manager selects a random sample of 10 representatives and determines the number of sales calls each representative made last month and the number of copiers sold.

LO1

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The Coefficient of Correlation, r

It shows the direction and strength of the linear relationship between two interval or ratio-scale variables It can range from -1.00 to +1.00. Values of -1.00 or +1.00 indicate perfect and strong correlation. Values close to 0.0 indicate weak correlation. Negative values indicate an inverse relationship and positive values indicate a direct relationship.

The Coefficient of Correlation (r) is a measure of the strength of the relationship between two variables.The Coefficient of Correlation (r) is a measure of the strength of the relationship between two variables.

LO2 Calculate, test, and interpret the relationship between two variables using the correlation coefficient.

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Correlation Coefficient - Interpretation

LO2

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Correlation Coefficient - Example

Using the Copier Sales of America data which a scatterplot is shown below, compute the correlation coefficient and coefficient of determination.

Using the formula:

LO2

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Correlation Coefficient - Example

What does correlation of 0.759 mean? First, it is positive, so we see there is a direct relationship between the number of sales calls and the number of copiers sold. The value of 0.759 is fairly close to 1.00, so we conclude that the association is strong.

However, does this mean that more sales calls cause more sales? No, we have not demonstrated cause and effect here, only that the two variables—sales calls and copiers sold—are related.

LO2

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Testing the Significance ofthe Correlation Coefficient

H0: = 0 (the correlation in the population is 0)H1: ≠ 0 (the correlation in the population is not 0)

Reject H0 if:t > t/2,n-2 or t < -t/2,n-2

LO2

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Testing the Significance ofthe Correlation Coefficient – Copier Sales Example

H0: = 0 (the correlation in the population is 0)

H1: ≠ 0 (the correlation in the population is not 0)

Reject H0 if:

t > t/2,n-2 or t < -t/2,n-2

t > t0.025,8 or t < -t0.025,8

t > 2.306 or t < -2.306

LO2

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Testing the Significance ofthe Correlation Coefficient – Copier Sales Example

The computed t (3.297) is within the rejection region, therefore, we will reject H0. This means the correlation in the population is not zero. From a practical standpoint, it indicates to the sales manager that there is correlation with respect to the number of sales calls made and the number of copiers sold in the population of salespeople.

Computing t, we get

LO2

13-12

Minitab Scatter Plots

No correlationNo correlation

Weak negativecorrelationWeak negativecorrelation

Strong positivecorrelationStrong positivecorrelation

LO2

13-13

Regression AnalysisIn regression analysis we use the independent variable (X) to estimate the dependent variable (Y).

The relationship between the variables is linear. Both variables must be at least interval scale. The least squares criterion is used to determine the

equation.

REGRESSION EQUATION An equation that expresses the linear relationship between two variables.REGRESSION EQUATION An equation that expresses the linear relationship between two variables.

LEAST SQUARES PRINCIPLE Determining a regression equation by minimizing the sum of the squares of the vertical distances between the actual Y values and the predicted values of Y.

LEAST SQUARES PRINCIPLE Determining a regression equation by minimizing the sum of the squares of the vertical distances between the actual Y values and the predicted values of Y.

LO3 Apply regression analysis to estimate the linear relationship between two variables.

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Linear Regression Model

LO3

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Regression Analysis – Least Squares Principle

The least squares principle is used to obtain a and b.

LO3

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Computing the Slope of the Line and the Y-intercept

LO3

13-17

Regression Equation - Example

Recall the example involving Copier Sales of America. The sales manager gathered information on the number of sales calls made and the number of copiers sold for a random sample of 10 sales representatives. Use the least squares method to determine a linear equation to express the relationship between the two variables. What is the expected number of copiers sold by a representative who made 20 calls?

LO3

13-18

Finding and Fitting the Regression Equation - Example

6316.42

)20(1842.19476.18

1842.19476.18

:isequation regression The

^

^

^

^

Y

Y

XY

bXaY

Step 1 – Find the slope (b) of the line

Step 2 – Find the y-intercept (a)

LO4 Interpret the regression analysis.

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Testing the Significance ofthe Slope – Copier Sales Example

H0: β = 0 (the slope of the linear model is 0)

H1: β ≠ 0 (the slope of the linear model is not 0)

Reject H0 if:

t > t/2,n-2 or t < -t/2,n-2

t > t0.025,8 or t < -t0.025,8

t > 2.306 or t < -2.306

LO5 Evaluate the significance of the slope of the regression equation.

13-20

Testing the Significance ofthe Slope – Copier Sales Example

Compute the t statistic and make a conclusion:

LO5

297335910

0184210.

..

bs

bt

Conclusion: The slope of the equation is significantly different from zero.

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The Standard Error of Estimate The standard error of estimate measures the

scatter, or dispersion, of the observed values around the line of regression

Formulas used to compute the standard error:

2

2

.

n

XYbYaYs xy

2

)( 2^

.

n

YYs xy

LO5

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Standard Error of the Estimate - Example

Recall the example involving Copier Sales of America. The sales manager determined the least squares regression equation is given below.

Determine the standard error of estimate as a measure of how well the values fit the regression line.

XY 1842.19476.18^

901.9210

211.784

2

)( 2^

.

n

YYs xy

LO5

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Standard Error of the Estimate - Excel

LO5

13-24

Coefficient of Determination

It ranges from 0 to 1. It does not give any information on the direction of

the relationship between the variables.

The coefficient of determination (r2) is the proportion of the total variation in the dependent variable (Y) that is explained or accounted for by the variation in the independent variable (X). It is the square of the coefficient of correlation.

The coefficient of determination (r2) is the proportion of the total variation in the dependent variable (Y) that is explained or accounted for by the variation in the independent variable (X). It is the square of the coefficient of correlation.

LO6 Calculate and interpret the Coefficient of Determination.

13-25

Coefficient of Determination (r2) – Copier Sales Example

•The coefficient of determination, r2 ,is 0.576, found by (0.759)2

•This is a proportion or a percent; we can say that 57.6 percent of the variation in the number of copiers sold is explained, or accounted for, by the variation in the number of sales calls.

LO6

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Computing the Estimates of Y

Step 1 – Using the regression equation, substitute the value of each X to solve for the estimated sales

6316.42

)20(1842.19476.18

1842.19476.18

Keller Tom

^

^

^

Y

Y

XY

LO7 Calculate and interpret confidence and interval estimates.

13-27

Computing the Estimates of Y

Step 1 – Using the regression equation, substitute the value of each X to solve for the estimated sales

4736.54

)30(1842.19476.18

1842.19476.18

Jones Soni

^

^

^

Y

Y

XY

LO7 Calculate and interpret confidence and interval estimates.

13-28

Plotting the Estimated and the Actual Y’s

LO7

13-29

Assumptions Underlying Linear Regression

For each value of X, there is a group of Y values, and these Y values are normally distributed. The means of these normal

distributions of Y values all lie on the straight line of regression. The standard deviations of these normal distributions are

equal. The Y values are statistically independent. This means that in

the selection of a sample, the Y values chosen for a particular X value do not depend on the Y values for any other X values.

LO7

13-30

Confidence Interval and Prediction Interval Estimates of Y

• A confidence interval reports the mean value of Y for a given X. • A prediction interval reports the range of values of Y for a particular value of X.

LO7

13-31

Confidence Interval Estimate - Example

We return to the Copier Sales of America illustration. Determine a 95 percent confidence interval for all sales representatives who make 25 calls.

LO7

13-32

Confidence Interval Estimate - Example

Step 1 – Compute the point estimate of Y

In other words, determine the number of copiers we expect a sales representative to sell if he or she makes 25 calls.

5526.48

)25(1842.19476.18

1842.19476.18

:isequation regression The

^

^

^

Y

Y

XY

LO7

13-33

Confidence Interval Estimate - Example

Step 2 – Find the value of t First, find the number of degrees of freedom.

degrees of freedom = n – 2 = 10 – 2 = 8. We set the confidence level at 95 percent. To find

the value of t, move down the left-hand column of Appendix B.2 to 8 degrees of freedom, then move across to the column with the 95 percent level of confidence.

The value of t is 2.306.

LO7

13-34

Confidence Interval Estimate - Example

2XX

2

XXStep 3 – Compute and 2XX

LO7

13-35

Confidence Interval Estimate - Example

Step 4 – Use the formula above by substituting the numbers computed in previous slides

Thus, the 95 percent confidence interval for the average sales of all sales representatives who make 25 calls is from 40.9170 up to 56.1882 copiers.

LO7

13-36

Prediction Interval Estimate - Example

We return to the Copier Sales of America illustration. Determine a 95 percent prediction interval for Sheila Baker, a West Coast sales representative who made 25 calls.

LO7

13-37

Prediction Interval Estimate - Example

Step 1 – Compute the point estimate of YDetermine the number of copiers we expect a sales representative to sell if he or she makes 25 calls.

5526.48

)25(1842.19476.18

1842.19476.18

:isequation regression The

^

^

^

Y

Y

XY

LO7

13-38

Prediction Interval Estimate - Example

Step 2 – Using the information computed earlier in the confidence interval estimation example, use the formula above.

If Sheila Baker makes 25 sales calls, the number of copiers she will sell will be between about 24 and 73 copiers.

LO7

13-39

Confidence and Prediction Intervals – Minitab Illustration

LO7

13-40